GIVEN: a > 0, a constant; is the finite region enclosed by AOAB. ΔΟΑΒ: 0 = (0,0), A = (a,0), B = (a, a) c is the closed path from A to B to O back to A. JΩ = ДОАВ, (The boundary of 2 is the triangle.) and given the 2D vector field F = (y³, 2xy²). EVALUATE: The circulation: F.ds 1 1 1 1 1 O 1 1 1 ' ΔΟΑΒ = 3Ω HINT: Use Green's theorem 1 B A F 1 X (y³, 2xy²)
GIVEN: a > 0, a constant; is the finite region enclosed by AOAB. ΔΟΑΒ: 0 = (0,0), A = (a,0), B = (a, a) c is the closed path from A to B to O back to A. JΩ = ДОАВ, (The boundary of 2 is the triangle.) and given the 2D vector field F = (y³, 2xy²). EVALUATE: The circulation: F.ds 1 1 1 1 1 O 1 1 1 ' ΔΟΑΒ = 3Ω HINT: Use Green's theorem 1 B A F 1 X (y³, 2xy²)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:GIVEN: a > 0, a constant;
Q is the finite region enclosed by
ΔΟΑΒ.
▲OAB:
O
=
(0,0), A = (a,0), B = (a, a)
c is the closed path
from A to B to O back to A.
ΘΩ = ΔΟΑΒ,
(The boundary of Q2 is the triangle.)
and given the 2D vector field
F = (y³, 2xy²).
EVALUATE: The circulation: F•ds
O
ΔΟΑΒ = 3Ω
HINT: Use Green's theorem
1
1
1
1
1
B
A
1
1
1
1
1
X
F = (v³, 2xy^²)
![[15] (1) GIVEN: a>0, a constant.
Consider the field F: R² R²,
F = (xy²-2y², x²y + y²)
and consider the path, c, once around the triangular region,
A =
(0,0)
(a,0),
3Ω = Δ ABC,
{Β
C = (a, a)
in the counter
clockwise direction.
EVALUATE: The circulation integral: F.ds
HINT: Use Green's theorem.
[Fote=625-34 lety
деду
Order of integration
must agree with the
functions in the limits,
of integration.
1
=
=
= 4
a
= 2
Try dy
a
2
2 √ ² x ² x
√2xy - (2xy-Ay) Sady
3
= ¾a³\
4√2+2x4, 5= {(57)[05752}
dx dy,
2
y x
NOTE: If you chose:
F = (xy²-2y², x²y + x²)
Then, F.ds=
F
y dy dx
a
↑
D
=
4
= 2
1.
↑ ↑ ↑
↑
1 ↑↑↑↑↑↑ 1
(xy² - 2y², x²y + y²)
a
√ ² [ 2³ | ²4
²
O
-box must go from
a function of x, to a higher
function of x.
NOTE: I made a "typo" when I wrote
the given field F a second time. So I gave full
crdit to whichever one you chose.
I also noticed that some of you
are not making your order of integration
consistent with your limits of integration.
Be careful about this.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4295bf62-da14-49f1-b3bf-6e678bc68de3%2Fc8b9f3d3-58ac-4707-ac79-97cb2b21dd17%2F7we44ig_processed.jpeg&w=3840&q=75)
Transcribed Image Text:[15] (1) GIVEN: a>0, a constant.
Consider the field F: R² R²,
F = (xy²-2y², x²y + y²)
and consider the path, c, once around the triangular region,
A =
(0,0)
(a,0),
3Ω = Δ ABC,
{Β
C = (a, a)
in the counter
clockwise direction.
EVALUATE: The circulation integral: F.ds
HINT: Use Green's theorem.
[Fote=625-34 lety
деду
Order of integration
must agree with the
functions in the limits,
of integration.
1
=
=
= 4
a
= 2
Try dy
a
2
2 √ ² x ² x
√2xy - (2xy-Ay) Sady
3
= ¾a³\
4√2+2x4, 5= {(57)[05752}
dx dy,
2
y x
NOTE: If you chose:
F = (xy²-2y², x²y + x²)
Then, F.ds=
F
y dy dx
a
↑
D
=
4
= 2
1.
↑ ↑ ↑
↑
1 ↑↑↑↑↑↑ 1
(xy² - 2y², x²y + y²)
a
√ ² [ 2³ | ²4
²
O
-box must go from
a function of x, to a higher
function of x.
NOTE: I made a "typo" when I wrote
the given field F a second time. So I gave full
crdit to whichever one you chose.
I also noticed that some of you
are not making your order of integration
consistent with your limits of integration.
Be careful about this.
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